Citation: Biss, Daniel K. "A generalized approach to the fundamental group." The American Mathematical Monthly 107.8 (2000): 711-720.
Web: https://www.jstor.org/stable/2695468
Tags: Mathematical, Philosophical
This paper considers the fundamental group itself as a topological space, endowed by the quotient topology on the compact open topology. In certain cases (like the Hawaiian earings) this group can fail to be discrete. The failure of the fundamental group to be discrete is exactly the failure of the group to have a universal cover, or stated otherwise, the failure of the group to be semi-locally simply connected.
In the case of non semi-locally simply connected spaces the theory of covering spaces breaks down. This paper argues that by considering the fundamental group as a topological space then some aspects of the theory of covering spaces can be recovered.
The important takeaway form this paper is that there is a natural topology on the fundamental group, and that this topology really does capture some universal features of your space. Importantly this topology does NOT make the fundamental group into a topological group. This paper claims that this is the case, but they are wrong. This is discussed at length in the following paper:
> Brazas, Jeremy. "The topological fundamental group and free topological groups." Topology and its Applications 158.6 (2011): 779-802.